Question

\(a)\) What does it mean for a set of operators to be functionally complete\(?\)

\(b)\)Is the set \(\{ {\bf{ + }}, \cdot \} \) functionally complete\(?\)

\(c)\)Are there sets of a single operator that are functionally complete\(?\)

Short answer

\(a)\)All Boolean expressions can be rewritten in an equivalent expression that uses only operators of set \(A\).

\(b)\)\(\{ {\bf{ + }}, \cdot \} \)is not functionally complete.

\(c)\)The set of operations is called functionally complete.

Step-by-step solution

Using the Boolean expression

(a)

If all Boolean expressions can be rewritten in an equivalent expression that uses only operators of set \(A\), then the set \(A\) of operations is said to be functionally complete.

Therefore, \(\{ {\bf{ + }}, \cdot \} \) is functionally complete.

Check whether it is functionally complete or not

(b)

If all Boolean expressions can be rewritten in an equivalent expression that uses only operators of set \(A\), then the set \(A\) of operations is said to be functionally complete.

Therefore, \(\{ {\bf{ + }}, \cdot \} \) is not functionally complete.

Check whether it is functionally complete or not

(c)

If all Boolean expressions can be rewritten in an equivalent expression that uses only operators of set \(A\), then the set \(A\) of operations is said to be functionally complete.

Therefore, \(\{ \uparrow \} \) is a single operator which is functionally complete.

So, a single operator can be functionally complete.